The discussion centers on the uniform convergence of the sequence of functions defined by fn(x) = 1/(nx+1) on the interval (0,1). It is established that fn(x) does not converge uniformly to the limit function f(x) = 0 due to the lack of continuity on a compact interval. The key argument involves demonstrating that for any chosen epsilon > 0, specifically epsilon = 1/4, there exists an x in (0,1) such that |fn(x) - f(x)| exceeds epsilon for infinitely many n. This confirms the non-uniform convergence of the sequence.
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snipez90 said:A sequence of functions does not converge to the limit function f if there exists some epsilon > 0 such that for infinitely many n, |f_n(x) - f(x)| > epsilon for some x in the domain of f_n. As you mentioned, f here is 0. Now pick epsilon to be say, 1/4. For arbitrary n, can you choose x so that f_n(x) > 1/4?